Problem: Factor the following expression: $x^2 - 12x + 32$
Solution: When we factor a polynomial, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + a)(x + b) &=& xx &+& xb + ax &+& ab \\ \\ &=& x^2 &+& {(a + b)}x &+& {ab} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + a)(x + b) }&\hphantom{=}&\hphantom{ xx }&\hphantom{+}&\hphantom{ (a + b)x }&\hphantom{+}& \\ &=& x^2 & & {-12}x& +& {32} \end{eqnarray} $ The coefficient on the $x$ term is $-12$ and the constant term is $32$ , so to reverse the steps above, we need to find two numbers that add up to $-12$ and multiply to $32$ You can try out different factors of $32$ to see if you can find two that satisfy both conditions. If you're stuck and can't think of any, you can also rewrite the conditions as a system of equations and try solving for $a$ and $b$ $ {a} + {b} = {-12}$ $ {a} \times {b} = {32}$ The two numbers $-8$ and $-4$ satisfy both conditions: $ {-8} + {-4} = {-12} $ $ {-8} \times {-4} = {32} $ So we can factor the expression as: $(x {-8})(x {-4})$